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When fractions are expressed in different bases, they can be terminating or repeating. For example, when $\frac{1}{5}$ is expressed in base $3,$ the result is
0.\overline{0121}_3 = 0.01210121 \dots,
which is repeating.

 

When \frac{1}{288} is expressed in base 25, is it terminating or repeating?

 Jun 18, 2024
 #1
avatar+1075 
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First, let's convert each number in the fraction to base 25. 

1 always stays the same. We want to convert 288 into base 25. We get that

\((BD)_{25} = (11 × 25^1) + (13 × 25^0) = (288)_{10}\)

 

So now, we have that \(\frac{1}{288}_{10} = \frac{1}{BD}_{25}\)

Doing some base division, we find that \(\frac{1}{BD}_{25} = 0.\overline{0.02468ACEGIKN}\)

 

As shown, the decimals keep repeating, so 1/288 in base 25 is a repeating decimal. 

 

So our answer is repeating. 

 

Thanks! :)

 Jun 18, 2024

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